Impulsive Dynamical Systems Research Articles

Robust multi-tracking of heterogeneous multi-agent systems with uncertain nonlinearities and disturbances

A robust multi-tracking problem is solved for heterogeneous multi-agent systems with uncertain nonlinearities and disturbances. The nonlinear function satisfies a Lipschitz condition with a time-varying gain, the integral of which is bounded by a linear function. A distributed impulsive protocol is proposed, where the position data and velocity data of desired trajectories are needed only at sampling instants. Based on the system decomposition technique, the error dynamic system of achieving multi-tracking is decomposed into two impulsive dynamic systems with vanishing perturbation and nonvanishing perturbation, respectively. Constructing a nominal model, then the multi-tracking problem is converted into the stability of impulsive dynamic system with nonvanishing perturbation under some conditions. It is proved that the proposed impulsive protocol is robust enough to solve the multi-tracking problem. Numerical examples are presented to illustrate the effectiveness of our theoretical results.

Dynamics of cholera epidemics with impulsive vaccination and disinfection

Waterborne diseases have a tremendous influence on human life. The contaminated drinking water causes water-borne disease like cholera. Pulse vaccination is an important and effective strategy for the elimination of infectious diseases. A waterborne disease like cholera can also be controlled by using impulse technique. In this paper, we have proposed a delayed SEIRB epidemic model with impulsive vaccination and disinfection. We have studied the pulse vaccination strategy and sanitation to control the cholera disease. The existence and stability of the disease-free and endemic periodic solution are investigated both analytically and numerically. It is shown that there exists an infection-free periodic solution, using the impulsive dynamical system defined by the stroboscopic map. It is observed that the infection-free periodic solution is globally attractive when the impulse period is less than some critical value. From the analysis of the model, we have obtained a sufficient condition for the permanence of the epidemic with pulse vaccination. The main highlight of this paper is to introduce impulse technique along with latent period into the SEIRB epidemic model to investigate the role of pulse vaccination and disinfection on the dynamics of the cholera epidemics.

Approximate controllability of impulsive non-local non-linear fractional dynamical systems and optimal control

We establish existence, approximate controllability and optimal control of a class of impulsive non-local non-linear fractional dynamical systems in Banach spaces. We use fractional calculus, sectorial operators and Krasnoselskii fixed point theorems for the main results. Approximate controllability results are discussed with respect to the inhomogeneous non-linear part. Moreover, we prove existence results of optimal pairs of corresponding fractional control systems with a Bolza cost functional.

The periodic solutions of the impulsive state feedback dynamical system

This work reviews some recent advances on the periodic solution of the semi-continuous dynamical system, which consists of two parts: the stability of periodic solution, the homoclinic and heteroclinic bifurcations. In the first part, the order-1 periodic solution is classified into three types at first. Then for type 1 periodic solution, by means of square approximation and a series of switched systems, the periodic solution is approximated by a series of continuous hybrid limit cycles. Hence, a general stability criteria are obtained by the method of successor function similar to the analysis in the ordinary differential equation. In the second part, the homoclinic and heteroclinic cycles are found for some specific parameter value in the prey-predator system. When the parameter varies, the cycles disappear and the system bifurcates an unique order-1 periodic solution. The geometry theory and the successor function are applied to obtain these bifurcations. Finally, we discuss some possible future trends in the periodic solution of the semi-continuous dynamical systems.

Fixed-time stabilization of impulsive Cohen–Grossberg BAM neural networks

This article is concerned with the fixed-time stabilization for impulsive Cohen–Grossberg BAM neural networks via two different controllers. By using a novel constructive approach based on some comparison techniques for differential inequalities, an improvement theorem of fixed-time stability for impulsive dynamical systems is established. In addition, based on the fixed-time stability theorem of impulsive dynamical systems, two different control protocols are designed to ensure the fixed-time stabilization of impulsive Cohen–Grossberg BAM neural networks, which include and extend the earlier works. Finally, two simulations examples are provided to illustrate the validity of the proposed theoretical results.

Hyers-Ulam stability of nonlinear impulsive Volterra integro-delay dynamic system on time scales

Hyers-Ulam stability of nonlinear impulsive Volterra integro-delay dynamic system on time scales

Fixed-time stability and stabilization of impulsive dynamical systems

This paper mainly tends to consider the fixed-time stability behavior for impulsive dynamical systems. An efficient theorem is established to construct an impulsive comparison system. By means of inequality analysis method, certain average impulsive interval and Lyapunov function, some sufficient conditions are given to ensure the fixed-time stability of impulsive dynamical systems. Moreover, as an important application, the fixed-time stabilization of a class of coupled impulsive neural networks is proposed. By designing a discontinuous control law, several new criteria are obtained to guarantee the fixed-time stabilization of the coupled impulsive neural networks. Finally, two numerical simulations are provided to illustrate the validity of the theoretical analysis.

Invariance and stability of global attractors for multi-valued impulsive dynamical systems

In this paper, we consider infinite-dimensional impulsive dynamical systems generated by evolution problems with impulses at the moments of intersection of trajectories with given subset of the phase space. Uniqueness of the corresponding Cauchy problem is not assumed, so we use multi-valued semiflows to describe the long-time dynamics of such systems. Considering global attractor as a compact minimal uniformly attracting set of the corresponding multi-valued semiflow we propose a criterion of existence of such a set. Under additional mild assumptions, we prove the invariance of the global attractor despite the fact that our approach does not require the invariance property of the set in the definition of the global attractor. Moreover, we prove that the obtained global attractor is stable under perturbation of the right-hand part of the evolution problem. We also provide examples which show that, in general, the global attractor is not preserved under perturbation of impulsive parameters.

Event‐triggered adaptive backstepping control for parametric strict‐feedback nonlinear systems

SummaryThis paper proposes a novel adaptive backstepping control method for parametric strict‐feedback nonlinear systems with event‐sampled state and input vectors via impulsive dynamical systems tools. In the design procedure, both the parameter estimator and the controller are aperiodically updated only at the event‐sampled instants. An adaptive event sampling condition is designed to determine the event sampling instants. A positive lower bound on the minimal intersample time is provided to avoid Zeno behavior. The closed‐loop stability of the adaptive event‐triggered control system is rigorously proved via Lyapunov analysis for both the continuous and jump dynamics. Compared with the periodic updates in the traditional adaptive backstepping design, the proposed method can reduce the computation and the transmission cost. The effectiveness of the proposed method is illustrated using 2 simulation examples.

On the continuity of the function describing the times of meeting impulsive set and its application.

The properties of the limit sets of orbits of planar impulsive semi-dynamic system strictly depend on the continuity of the function, which describes the times of meeting impulsive sets. In this note, we will show a more realistic counter example on the continuity of this function which has been proven and widely used in impulsive dynamical system and applied in life sciences including population dynamics and disease control. Further, what extra condition should be added to guarantee the continuity of the function has been addressed generally, and then the applications and shortcomings have been discussed when using the properties of this function.

Stability analysis of linear impulsive delay dynamical systems via looped-functionals

This article investigates the asymptotic stability of impulsive delay dynamical systems (IDDS) by using the Lyapunov–Krasovskii method and looped-functionals. The proposed conditions reduce the conservatism of the results found in the literature by allowing the functionals to grow during both the continuous dynamics and the discrete dynamics. Sufficient conditions for asymptotic stability in the form of linear matrix inequalities (LMI) are provided for the case of impulsive delay dynamical systems with linear and time-invariant (LTI) base systems (non-impulsive actions). Several numerical examples illustrate the effectiveness of the method.

Nonlinear impulsive systems: 2D stability analysis approach

This paper contributes to the stability analysis for nonlinear impulsive dynamical systems based on a vector Lyapunov function and its divergence operator. The new method relies on a 2D time domain representation. Different types of stability notions for a class of nonlinear impulsive systems are studied using a vector Lyapunov function approach. The results are applied to analyze the stability of a class of Lipschitz nonlinear impulsive systems based on Linear Matrix Inequalities. Some numerical examples illustrate the feasibility of the proposed approach.

Exponentially Dissipative Control for Singular Impulsive Dynamical Systems

This paper studies the problem of exponentially dissipative control for singular impulsive dynamical systems. Some necessary and sufficient conditions for exponential dissipativity of such systems are established in terms of linear matrix inequalities (LMIs). A state feedback controller is designed to make the closed-loop system exponentially dissipative. A numerical example is given to illustrate the feasibility of the method.

Model-Based Adaptive Event-Triggered Control of Strict-Feedback Nonlinear Systems.

This paper is concerned with the adaptive event-triggered control problem of nonlinear continuous-time systems in strict-feedback form. By using the event-sampled neural network (NN) to approximate the unknown nonlinear function, an adaptive model and an associated event-triggered controller are designed by exploiting the backstepping method. In the proposed method, the feedback signals and the NN weights are aperiodically updated only when the event-triggered condition is violated. A positive lower bound on the minimum intersample time is guaranteed to avoid accumulation point. The closed-loop stability of the resulting nonlinear impulsive dynamical system is rigorously proved via Lyapunov analysis under an adaptive event sampling condition. In comparing with the traditional adaptive backstepping design with a fixed sample period, the event-triggered method samples the state and updates the NN weights only when it is necessary. Therefore, the number of transmissions can be significantly reduced. Finally, two simulation examples are presented to show the effectiveness of the proposed control method.

Poincaré recurrence theorem in impulsive systems

In this article, we generalize the Poincaré recurrence theorem to impulsive dynamical systems in $\mathbb R^n$. For a measure preserving system, we present some sufficient conditions to establish an impulsive system that is also measure preserving. Then, two recurrence theorems are proved. Finally, we use two examples to illustrate our results.

一类具有季节效应的脉冲控制生物系统动力学研究

基于生态学理论与数学生物学知识，在动态建模过程中加入了Hassell-Varley功能反应函数，建立了一类具有季节效应的脉冲控制生物动力系统。借助脉冲微分方程的Floquet定理与比较定理，分析了系统半平凡周期解的存在性、局部渐近稳定性和全局渐近稳定性，同时讨论了系统生物种群的灭绝性与持久生存性。这些研究结果为进一步研究如何运用脉冲控制策略维持生态种群持久生存提供了一定的理论支撑。 In this paper, firstly, on the basis of ecology theory and mathematical biology knowledge, an impulsive controlled biological dynamical system with seasonal effect has been established by introducing Hassell-Varley functional response in the process of dynamic modeling. Secondly, using the Floquet theory and comparison theorem of impulsive differential equations, the existence, local asymptotic stability and global asymptotic stability of the semi-trivial periodic solution have been analyzed, and then the extinction and permanence of biological populations in the system have also been discussed. Finally, all those results can provide some theoretical support for further researching how to utilize control strategy to maintain the survival of ecological populations.

Global attractors of impulsive parabolic inclusions

In this work we consider an impulsive multi-valued dynamical system generated by a parabolic inclusion with upper semicontinuous right-hand side \begin{document}$\varepsilon F(y)$\end{document} and with impulsive multi-valued perturbations. Moments of impulses are not fixed and defined by moments of intersection of solutions with some subset of the phase space. We prove that for sufficiently small value of the parameter \begin{document}$\varepsilon>0$\end{document} this system has a global attractor.

Exponential consensus of non‐linear stochastic multi‐agent systems with ROUs and RONs via impulsive pinning control

Exponential consensus of non-linear stochastic multi-agent systems is investigated. The authors consider a class of non-linear stochastic multi-agent systems that are subject to time-varying delays, randomly occurring uncertainties and randomly occurring non-linearities. Impulsive pinning control algorithms are proposed to ensure that follower agents track the leader under a fixed topology. On the basis of the Lyapunov function and the Halanay differential inequality of impulsive dynamical systems, we derive sufficient conditions for the globally exponential consensus of the multi-agent systems. Finally, numerical simulations demonstrate the effectiveness of the proposed theoretical results.

Attractors for impulsive non-autonomous dynamical systems and their relations

In this work, we deal with several different notions of attractors that may appear in the impulsive non-autonomous framework and we explore their relationships to obtain properties regarding the different scenarios of asymptotic dynamics, such as the cocycle attractor, the uniform attractor and the global attractor for the impulsive skew-product semiflow. Lastly, we illustrate our theory by exhibiting an example of a non-classical non-autonomous parabolic equation with subcritical nonlinearity and impulses.

Monotone impulsive dynamical systems

In this work, we introduce the concept of monotone impulsive dynamical systems. We exhibit sufficient conditions for a set to be Zhukovskij quasi stable in dissipative monotone impulsive systems. Also, some recursive properties as minimality and recurrence are related with monotone impulsive systems.